A Graph Signal Processing Perspective on Numerical Sequence Representations in LLM In-Context Learning

Pretrained large language models (LLMs) have demonstrated in-context learning (ICL) capabilities for numerical inference over sequences serialized as text. Prior work has identified and characterized this form of numerical inference primarily through output-level evaluations such as prediction error. However, how numerical information is organized within LLM representations remains much less understood. To study this internal organization, we adopt a graph signal processing perspective in which attention induces a weighted graph over tokens, while token hidden states define signals on its nodes. Quantitative graph-spectral diagnostics and qualitative token-graph visualizations reveal that representations become more clearly differentiated by input dynamical complexity as context length increases. Simpler inputs produce attention-induced token graphs with stronger global connectivity and smoother, spectrally concentrated hidden-state signals, whereas more complex inputs produce more localized graphs and hidden-state signals with broader spectral support and greater high-frequency energy. Together, these findings point to systematic, context-dependent internal signatures associated with numerical ICL that are conserved across model families.

Sheaf-theoretic Signal Processing on Graphs: Spectral Theory, Filtering, and Sampling

Modern sensing, communication, and learning systems generate heterogeneous network signals, with local data differing in dimension, modality, and geometric structure. Processing such data requires a mathematical framework capable of simultaneously modeling heterogeneous local signal spaces and the transformations relating them. Network sheaves provide such a framework by associating local vector spaces with network entities and linear restriction maps with their interactions. This is the first paper to develop a unified sheaf signal processing (SSP) framework on network sheaves, extending the fundamental operations of signal processing, namely spectral analysis, filtering, and sampling, to heterogeneous local spaces. Unlike graph and topological signal processing, where signals are modeled over a common vector space, SSP jointly models heterogeneous local signal spaces and the linear transformations relating neighboring spaces through restriction maps. We define the Sheaf Fourier Transform (SFT), whose frequencies quantify signal inconsistency induced by the network topology, the restriction maps, and the local geometry. Building on this representation, we develop polynomial sheaf filters and formulate sampling as the joint selection of network nodes and intra-node components. We derive perfect recovery conditions for bandlimited sheaf signals and propose a greedy sampling-set design algorithm. To incorporate application-dependent signal models, including different bases, dictionaries, and learned embeddings, we introduce representation sheaves and characterize the natural transformations that preserve spectral properties and guarantee interoperability across representations. Experiments on synthetic, motion-capture, and financial datasets validate the proposed framework and demonstrate consistent improvements over canonical graph signal processing baselines.

Scalable Graph Coreset Selection via Greedy Sampling

Sampling representative nodes from large graphs is fundamental to graph signal processing and network analysis, yet existing methods require access to the full graph Laplacian, making them impractical at scale. We propose a simple and effective column-selective graph sampling algorithm based on a minimum inner product greedy selection rule. At each iteration, the algorithm accesses only a small random subset of Laplacian columns, requiring no eigendecomposition or global graph traversal, making it well-suited for large-scale graphs where the full Laplacian cannot be stored in memory. We analyze the algorithm under the stochastic block model and show that, when the degree distribution is balanced across nodes, the algorithm achieves sampling proportional to cluster size, and that the resulting mean estimate is controlled for band-limited graph signals in the Paley-Wiener space, with the error decaying as inter-cluster connectivity weakens. Numerical experiments on both synthetic and real-world data validate the effectiveness of the proposed method.

Filter Learning for Subgraphs: Algebras and Performance Risk Bounds

Graph signal processing tasks that leverage spectral information typically assume access to the complete graph topology, which is often unavailable in practice. We propose a systematic framework for subgraph filter learning (SFL), where subgraph-supported operators approximate ambient graph filters under partial observations. We formulate SFL as a statistical learning problem in which optimal subgraph operators are inherently data-dependent. To address the difficulty of directly estimating such operators, we develop a subgraph filter algebra based on distance-aware Laplacian constructions, defining a structured and controllable class of filters for effective approximation. We further establish performance risk bounds under the least squares loss, quantifying how well the learned operator approximates the restricted ambient mapping. Experiments real-world datasets show that, for SFL tasks, the proposed algebraic models consistently outperform polynomial filters, distribution-agnostic operators, and direct numerical filter learning baselines that attempt to recover the underlying structure from data.

Connectivity Estimation using Stochastic Graph Heat Modelling

A growing number of techniques leverage the spatial structures that underlie many real-world datasets. Despite these advances, the complementary task of estimating spatial structures and understanding their role within these techniques has often been overlooked. In neurophysiological data analysis specifically, numerous methods exist to estimate brain connectivity, but most are not explicitly model-based, dynamic, multivariate, or directed. To address these limitations, we previously introduced noise-driven heat modelling on graphs for neurophysiological connectivity estimation. In this study, we extend this framework by relaxing earlier noise assumptions and adding regularisation to improve robustness. We also develop a simulation procedure to characterise and evaluate our technique in a controlled setting. Finally, we demonstrate that the technique is able to capture meaningful spatial structure across two experiments, each using two real-world datasets. The explicit model formulation of our connectivity estimator has the potential to improve the interpretability of graph-based techniques across a wide range of applications. The code implementing our method is available at https://github.com/sgoerttler/Heat_Connectivity.

Directed Graph Topology Inference via Graph Filter Identification

We address the problem of inferring a directed network from nodal measurements generated by linear diffusion dynamics on the sought graph. Observations are modeled as the outputs of a graph convolutional filter, i.e., a polynomial (with unknown coefficients) of a local diffusion graph-shift operator encoding the latent graph topology, excited with an ensemble of independent graph signals with arbitrarily-correlated nodal components. Unlike prior efforts that considered undirected graphs and white signal excitations, here the graph-shift operator and the observations' covariance matrix are not simultaneously diagonalizable. In this challenging context, we first rely on measurements of the output signals along with prior statistical information on the inputs to identify the diffusion filter. Such system identification problem involves solving a system of quadratic matrix equations, which we show is identifiable under spectral-diversity assumptions on the input covariances. For algorithmic purposes we recast it as a smooth quadratic minimization subject to Stiefel manifold constraints. Subsequent identification of the network topology given the graph filter estimate boils down to finding a sparse and structurally admissible shift that commutes with the given filter, thus, forcing the latter to be a polynomial in the sought graph-shift operator. A joint graph filter and topology identification algorithm is also proposed, which alternates between the aforementioned steps in a mutually reinforcing fashion to offer improved sample complexity. Numerical tests corroborate the effectiveness of the proposed algorithms in recovering synthetic digraphs and real-data case studies, and illustrate their potential utility on urban mobility analyses as well as portfolio optimization.

Low-rank Updates in Slowly Time-varying Graphs for Spatial-Temporal Signal Interpolation

A crucial assumption in graph signal processing (GSP) is the existence of an underlying graph that captures the pairwise similarities between nodes, allowing filters to be designed based on this graph for tasks such as denoising. For spatial-temporal data in which node-to-node similarities evolve over time, a static spatial graph is insufficient. In this paper, to represent slowly time-varying pairwise relationships, we model the graph changes in two consecutive adjacency matrices $P = W^{(2)} - W^{(1)}$ across time as a low-rank matrix. % Specifically, given an initial adjacency matrix $W^{(1)}$ at time $t=1$, we jointly interpolate a signal $x_2$ and estimate $W^{(2)}$ at $t=2$ using both a graph signal smoothness prior for $x_2$ and a low-rank prior on $¶$. We alternate optimization steps. With $W^{(2)}$ fixed, $x_2$ is interpolated by solving a linear system. Alternatively, holding $x_2$ fixed, $W^{(2)}$ is updated via proximal gradient descent (PGD). The proximal mapping of the rank term $Gamma(W^{(2)} - W^{(1)})$ is approximated in linear time using a fast orthogonal matching pursuit (OMP) algorithm that selects a sparse combination of atoms from a dictionary $cR$ formed by the outer products of $W^{(1)}$'s eigenvectors. We unroll iterations of our algorithm into layers to build a lightweight neural network for limited data-driven parameter tuning. Experiments show that our joint optimization achieves better signal interpolation compared to existing time-varying graph models.